ME111 Instructor: Peter Pinsky Class #21 November 13, 2000

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1 Toda s Topics ME Instructor: Peter Pinsk Class # November,. Consider two designs of a lug wrench for an automobile: (a) single ended, (b) double ended. The distance between points A and B is in. and the handle diameter is.65 in. What is the maimum force possible before ielding the handle if 45 kpsi? Failure of ductile materials under static loading. The von Mises ield criterion. The maimum shear stress criterion.. A storage rack is to be designed to hold a roll of industrial paper. The weight of the roll is 5.9 kn, and the length of the mandrel is.65 m. Determine suitable dimensions for a and b to provide a factor of safet of.5 if: (a) The beam is a ductile material with Mpa (b) The beam is a brittle material with ut 5 Mpa, uc 57 Mpa. Reading Assignment Juvinall, For the problem in Eample 9. (torsion-bar spring), what diameter d will provide a factor of safet of N against ielding based on von Mises with 5 MPa. Problem et #7. Due Wednesda, November.. The stress (in kpsi) at a point is given b,, Calculate the factor of safet against failure if the material is: (a) Brittle with ut 5, uc 9 and using modified-mohr theor. (b) Ductile with 4 using (i) the ma. shear stress criterion, and (ii) von Mises criterion. 5. Repeat Eample. (a) and (b) using the maimum shear stress criterion. 6. For the beam shown, determine the factor of safet for: (a) Ductile material with Mpa (b) Brittle material with ut 5 Mpa, uc 57 Mpa.

2 . Failure of Ductile Materials Under tatic Loading tatic Loads: Brittle materials fail b cracking or crushing and are tpicall limited b their tensile strengths Ductile materials fail b ielding and are tpicall limited b their shear strengths Recall Uniaial Tension Test: Problem F b F d F d Problem F d, F b Problem 6 F d

3 . Three-Dimensional tress tates Recall Torsion Test: It s useful to think about -d stress states using principal stresses: F b F d Principal stresses I + I I F d I I I F b Maimum shear stresses ma(,, ) ma

4 . plit of tress into Mean and Distortional Components.4 Yielding of Ductile Materials A ductile material ields when a ield criterion is eceeded. m m m m Given principal stress state m ( + + ) Mean stress produces volumetric strain Two ield criteria are important: The von Mises Yield Theor The strain energ per unit volume of an elastic bod has the form: U U m + U ν Um 6E + ν Ud E U h d [ ( + + )] [ + + ] where is the strain energ per unit volume associated with pure volume change (dilation) due to the mean (hdrostatic) stress: m ( + + ) and where U is the strain energ per unit volume associated with pure distortion d Note that in a tensile test at ield: m m Produces distortional strain responsible for plastic ielding, and, in this special case, U E + ν d 4

5 The von Mises ield criterion predicts failure in a general -d stress state when the distortion energ per unit volume U d is equal to the distortion energ per unit volume in the tensile test specimen at failure, i.e. U + ν E d + ν ν [ + + ] + E E Von Mises ellipse (plane stress) Yield surface No ielding Pure tension Pure shear (torsion) or, + + In terms of, and stresses ( ) + ( ) + ( ) + 6( + + ) pecial cases: Pure tension For plane stress, Note, this is rather arbitrar but we ll work with this as the plane stress state. + In terms of, and stresses Pure shear (torsion), ma ma + + 5

6 Von Mises Effective tress Convenient to introduce the von mises effective stress Eample. For the bracket shown, determine the von Mises stress and factor of safet against ielding at points (a) and (b) if,. + + (The von Mises effective stress is defined as the uniaial tensile stress that would create the same distortion energ U d as is created b the the actual combination of applied stress) von Mises (distortion energ) Yield Criterion The von Mises ield criterion predicts failure when: R, M A 5 R, M R, M From previous analsis we found: At (a): 7, 759 B C (a) (b) A Factor of afet Against Yielding N At (b): + N,. 7,79 596, 4 + N,.9 5,74 7,79 5,74 6

7 Eample. A thin-walled clindrical pressure vessel is subject to an aial force and torque loads as shown: (a) Given: r p 5MPa, r 5mm, t mm, p P T T 45kN mm, 9MPa Determine the range of values of the aial load P which will provide a factor of safet against ielding of at least.4 based on the von Mises criterion. (b) Given: p MPa, t 5mm, T 8kN mm, P kn, 9MPa Determine the range of values of the radius r which will provide a factor of safet against ielding of at least.4 based on the von Mises criterion. t.5 Maimum hear tress Criterion (Tresca Condition) Another important criteria is based on the theor that shear stress controls ielding (in contrast to von Mises theor based on distortion energ). This theor was developed before the von Mises criterion and in practice is slightl more conservative. The theor is eas to use in an analtical setting but is not well suited to use in finite element codes because of the corners in the ield surface (see below). The maimum shear stress theor states that ielding will occur when the maimum shear stress reaches the shear stress in a uniaial test specimen at ield, i.e. or ma(,, ) ma(,, ) For plane stress with ma(,, ) Factor of afet Against Yielding ma(,, ) N Remark: von Mises is the preferred theor 7

8 Von Mises and Tresca for Plane tress Eample. For the bracket shown, determine the factor of safet against ielding using the maimum shear stress theor at points (a) and (b) if,. tress states inside the ield surface have not ielded Von Mises ield surface Tresca ield surface von Mises: Tresca: Pure tension Pure shear (torsion) R, M A 5 R, M R, M From previous analsis we found: At (a): 7, 759 B C (a) (b) A von Mises: Tresca:, N.7 ma(,, ) 7,86 At (b): 596, 4, N.84 ma(,, ) 5,49 8